Looking for information on the WEB, we find an interesting work by José Ramón Gregorio Guirles on how to work with calculus in the first cycle of primary school, which we want to share.
When we speak of calculus we are referring to two different types of mathematical questions and works.
- To the activities and problem situations specifically destined to the understanding and construction of the concepts of the basic operations of this cycle, and to the flexibility and number sense related to them.
- To the different modalities and tools that students can use to carry out the operations of addition and subtraction typical of this cycle. We are talking about mental calculations, calculator calculations and calculations made with pencil and paper using the algorithms of addition and subtraction.
It is evident that, from the beginning, calculation work (in any of its modalities) must be linked to problem solving, since they are still tools that acquire their meaning and real dimension when they serve for it (true sense of know how to operate).
Some basic work considerations related to calculation:
- The first activities and calculation problems should serve to give meaning to the operations of adding and subtracting. In this first phase of work, the resolution procedures will be the manipulation and counting with chips, chickpeas, …, the use of materials that symbolize numbers (letters, drawings, graphic problems, …), and the use of the first and simplest strategies of mental calculation. Through them, students must build the concepts of adding and subtracting and begin to solve problems. We are not talking about the algorithm; Knowing how to add and subtract is not the same as knowing how to add and subtract. Knowing how to add and subtract means knowing, among other things, when to use each operation and identifying problematic situations that can be resolved with one operation or another. This is the true conceptual understanding;
- In a second phase, we must work on problems and activities aimed at the construction and mastery of other basic calculation strategies and the automation of addition and subtraction tables (automatic and reflective mental calculation).
- The third phase will be the specific work around the algorithms of addition and subtraction (calculations with pencil and paper). We will only have to proceed to learn the algorithms of addition and subtraction when the students have understood what it means to add and subtract, and when they have a good mental numerical command: decompositions of numbers in different ways, a certain command of addition tables and subtraction, number sense and mental flexibility in calculations (mastery of different strategies).
Learning the addition and subtraction algorithms must be based on understanding. Therefore, it must be a learning path in which different mental and written solving strategies are used, and in which students have the opportunity to investigate and construct different ways (algorithms) to perform addition and subtraction, before get to the academic algorithms, which are the last stage of this journey through different strategies.
At this level of algorithmic resolution, it is convenient to work first with the addition algorithm and then that of subtraction. The strategies and algorithms learned for addition will have a positive transfer to use different strategies and algorithms to subtract.
- The calculator can be used at any time, and from the beginning: to reinforce automatic mental calculations, to do research and reach numerical and operational conclusions (number sense), to support the construction of numerical concepts and operations, to facilitate exploration and problem solving, to give them greater autonomy and confidence, …
Let’s think that the calculator is still just another tool that, when used well, should be at the service of understanding and solving mathematical problems. Good use of the calculator involves thinking and knowing what to do with it. Thus, the question of calculation is not so much a problem of working the different types of calculation separately (one day mental calculation, another written, another strategies, another …), but a work of integration and didactic progression of activities. In this progression, understanding and mental mathematics are priorities (numerical decompositions, flexibility and number sense, mental addition and subtraction, problems with mental resolution or with a calculator).
This is the most important process, the one that guarantees true mathematical literacy: thinking, understanding, speculating, building knowledge, … Skipping over this process, focusing all mathematical efforts and time on training students in how to do addition operations and subtraction, leaving aside the understanding of numbers and operations, means in most cases to get functional illiterate students in mathematics.